Complexity reduction in low-delay Farrowstructure-based. filters utilizing linear-phase subfilters

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1 Coplexity reduction in low-delay Farrowstructure-based variable fractional delay FIR filters utilizing linear-phase subfilters Air Eghbali and Håkan Johansson Linköping University Post Print N.B.: When citing this work, cite the original article. IEEE. Personal use of this aterial is peritted. However, perission to reprint/republish this aterial for advertising or prootional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted coponent of this work in other works ust be obtained fro the IEEE. Air Eghbali and Håkan Johansson, Coplexity reduction in low-delay Farrow-structurebased variable fractional delay FIR filters utilizing linear-phase subfilters,, Eur. Conf. Circuit Theory Design, IEEE. Postprint available at: Linköping University Electronic Press

2 th European Conference on Circuit Theory and Design (ECCTD) Coplexity Reduction in Low-Delay Farrow-Structure-Based Variable Fractional Delay FIR Filters Utilizing Linear-Phase Subfilters Air Eghbali and Håkan Johansson Division of Electronics Systes, Departent of Electrical Engineering, Linköping University, Sweden. E-ail: Abstract This paper proposes a ethod to design low-delay fractional delay (FD) filters, using the Farrow structure. The proposed ethod eploys both linear-phase and nonlinearphase finite-length ipulse response(fir) subfilters. This is in contrast to conventional ethods that utilize only nonlinear-phase FIR subfilters. Two design cases are considered. The first case uses nonlinear-phase FIR filters in all branches of the Farrow structure. The second case uses linear-phase FIR filters in every second branch. These branches have ilder restrictions on the approxiation error. Therefore, even with a reduced order, for these linear-phase FIR filters, the approxiation error is not affected. However, the arithetic coplexity, in ters of the nuber of distinct ultiplications, is reduced by an average of 3%. Design exaples illustrate the ethod. Index Ters Farrow Structure, Low-Delay, Fractional Delay, Low-Coplexity. I. INTRODUCTION Applications such as edical signal processing [], digital counications [], tie/delay estiation [3], and nonunifor sapling [4] require either to perfor sapling rate conversion (SRC) or to delay a digital signal by fractions of the sapling period. A software defined radio ay require both of these [5]. For exaple, one ay perfor SRC and also estiate the phase/carrier. In ultistandard receivers, there is a need to dynaically perfor SRC and signal delaying so as to support various standards. We can use dedicated blocks to perfor SRC and signal delaying, for each standard, but this requires to either (i) design a large set of filters offline, or (ii) design the filters online. An efficient way, to solve the above proble, is to use the Farrow structure [6]. The Farrow structure is designed to approxiate an allpass function with an adjustable fractional delay (FD) of µ(n). The FD value can change during each sapling period. For SRC, one should delay each input saple by a specific µ(n) whereas for signal delaying, all input saples are delayed by a fixed µ(n). For siplicity, the rest of the paper uses the ter µ instead of µ(n). A. Contribution of the Paper The Farrow structure, shown in Fig., is generally coposed of L + fixed finite-length ipulse response (FIR) subfilters S k (z) and variable ultipliersµ. By a proper design of S k (z), the Farrow structure can approxiate FD filters with an adjustable µ. With realizable (nonideal) filters, the resulting FD filters will have an approxiation error which can be reduced as in, e.g., [7] []. The ajority of the earlier design ethods choose S k (z) to be linear-phase FIR filters. Although this reduces the arithetic coplexity, because of the resulting syetry or antisyetry in the ipulse responses of S k (z), the overall delay increases. This is partly due to the strict constraints which are inherently iposed on soe of S k (z) thereby increasing the values of N k. With odd N k, the subfilters in branches k =,,4,..., have a higher order as copared to other branches [8]. For even N k, the subfilter S (z) is a pure delay and, then, the subfilters in branches k =,3,..., have a higher order [8]. This paper proposes a ethod to design FD filters, with odd N k, in which the Farrow structure uses both nonlinear-phase and linear-phase FIR subfilters. In other words, we replace the high-order linear-phase FIR subfilters S k (z), k =,,..., with lower-order nonlinear-phase FIR filters. By eans of design exaples, we show that this reduces the arithetic coplexity, as opposed to the case with nonlinear-phase FIR filters in all branches. This paper ainly considers cases where the nonlinearphase subfilters have the sae orders in all branches. Also, the linear-phase subfilters have the sae orders. The design exaples show that we can indeed eet the specifications with a reduced arithetic coplexity. However, the paper also outlines soe design exaples to show that we can allow subfilters of different orders, in different branches, thereby further reducing the arithetic coplexity. This has not been treated earlier for low-delay FD filters. We consider iniax designs by iniizing the axiu of the odulus of the coplex error, i.e., the difference between the frequency responses of (i) the nonideal Farrow structure, and (ii) the ideal FD filter. Note that the proposed ethods can also be applied if N k is even. The subfilters can alternatively be infinite-length ipulse response (IIR) but this paper focuses on the FIR case //$6. IEEE

3 x(n) S L (z) S (z) S (z) S (z) y(n) Fig.. Farrow structure with fixed subfilters S k (z) and variable FD µ. TABLE I POSSIBLE TYPES OF LINEAR-PHASE FIR FILTER S k (z). B. Paper Outline N k k Type even even I even odd III odd even II odd odd IV Section II outlines the Farrow structure and the error frequency responses. In Section III, the filter design proble is treated and a discussion on the arithetic coplexity is provided. Further, soe design exaples are also illustrated. Finally, Section IV concludes the paper. II. FARROW STRUCTURE The Farrow structure is coposed of fixed general FIR subfilters S k (z), k =,,...,L, of order N k where the transfer function is [6] L H(z,µ) = S k (z)µ k, µ.5. () k= Here, µ is the FD value. If S k (z) are chosen to exhibit a linear phase, they could have any of the four types [3] of linear-phase FIR filters as in Table I. A. Error Frequency Responses Generally, S k (z), k =,,...,L, are designed so as to approxiate the following desired causal coplex and unwrapped phase responses H des (e jωt,µ) = e j( +µ)ωt, () Φ des (ωt,µ) = ( +µ)ωt (3) where ωt [,ω c T]. If all S k (z) are linear-phase FIR filters, the overall delay is defined as = ax k(n k ). (4) With linear-phase FIR subfilters, the value of can be intolerably large due to the strict constraints on soe of S k (z) [8]. If H(e jωt,µ) = H(e jωt,µ) e jφ(ωt,µ), (5) the coplex, agnitude, and phase errors are H e c(e jωt,µ) = H(e jωt,µ) e j( +µ)ωt, (6) H e (ωt,µ) = H(e jωt,µ), (7) H e p(ωt,µ) = Φ(ωT,µ)+( +µ)ωt. (8) With a known bound on H e c(e jωt,µ), the bounds on H e (ωt,µ) and H e p(ωt,µ) can be obtained [8]. III. FILTER DESIGN Soe applications ay require to have < ax k(n k ) which would further restrict S k (z) thereby increasing N k. With odd N k, this restriction ainly increases the orders of S k (z), k =,,... To alleviate this, one can increase the degrees of freedo, in the filter design, by using general nonlinear-phase FIR filters for all S k (z). Although this gives low-delay FD filters, it unnecessarily increases the arithetic coplexity. This paper shows that the arithetic coplexity can be reduced by allowing soe of S k (z) to be linear-phase and keeping the others as nonlinear-phase. With odd N k, this aounts to having nonlinear-phase FIR filters in branches k =,,..., and linear-phase FIR filters in branches k =,3,... In the exaples of this paper and for a low-delay specification, we use a new which is around 6% of that given by (4). For the sae approxiation error, this new can further be reduced by increasing N k. With a given and ω c T, we design two FD filters for coparison purposes. In the first one, all S k (z) are nonlinearphase FIR filters having the sae orders. For the second FD filter, we reduce the orders of S k (z), k =,3,..., and also replace the with Type IV linear-phase FIR filters. The filter design proble, considered here, is forulated as in δ subject to (9) H e c(ωt,µ) δ where ωt [,ω c T]. This design proble is convex and it can be solved using the standard filter design ethods eploying, for exaple, the real rotation theore. In this paper, we use the algorith infiniax to solve (9). In the exaples of Section III-B, we have the sae orders N k in branches k =,,... Siilarly, the subfilters in branches k =,3,..., have the sae orders. However, as we shall show in Section III-C, it is possible to obtain low-delay FD filters in which S k (z) have different orders in different branches. This can further reduce the arithetic coplexity. A. Arithetic Coplexity For coparison, we use the arithetic coplexity in ters of the nuber of distinct ultiplications, as ultipliers are ore costly to ipleent than adders. If all S k (z) are general nonlinear-phase FIR filters of orders N k, the ultiplicative coplexity of the Farrow structure is C g = L (N k +). () k= By replacing soe of S k (z) with Type IV linear-phase FIR filters, the ultiplicative coplexity becoes L k= C l = (N k +)+ L N k+ + k= odd L L k= (N k +)+ L k= N k+ + even L. ()

4 .5..5 =[ ] x =[ ] π.π.3π.4π.5π.6π.7π.8π =[ 7 7 ].5..5.π.π.3π.4π.5π.6π.7π.8π.π.π.3π.4π.5π x 3 =[ ].5.5.π.π.3π.4π.5π Fig.. Coplex error for low-delay FD filters with = 3.5. Fig. 3. Coplex error for low-delay FD filters with = 4.5. In other words, coplexity savings occur in the branches utilizing linear-phase FIR S k (z). This saving is due to (i) a saller N k, and (ii) the inherent antisyetry in the ipulse responses of Type IV linear-phase FIR filters. Note that the proposed ethod reduces the nuber of adders as well, but to a lower extent. B. Design Exaples This section outlines soe design exaples and the corresponding savings in the ultiplicative coplexity. In all of these exaples, the two FD filters eet practically the sae specifications. Exaple: If ω c T =.8π, = 3.5, and L = 4, the case with nonlinear-phase S k (z) and N k = {,,,,} gives δ.4. With nonlinear-phase S k (z), k =,,..., and linear-phase S k (z), k =,3,..., where N k = {, 7,, 7, }, the ultiplicative coplexity is reduced by 7%. The characteristics of these FD filters are shown in Fig.. Exaple : With ω c T =.5π, = 4.5, and L = 3, nonlinear-phase S k (z) with N k = {3,3,3,3} gives δ.. If S k (z), k =,,..., are nonlinear-phase and S k (z), k =,3,..., are linear-phase where N k = {3, 9, 3, 9}, the ultiplicative coplexity is 3% lower resulting in the filter charactersitics of Fig. 3. Exaple 3: For ω c T =.7π, =.5, L = 4, and nonlinear-phase S k (z) with N k = {9,9,9,9,9}, we have δ.6. The choice of nonlinear-phase S k (z), k =,,..., and linear-phase S k (z), k =,3,..., where N k = {9,5,9,5,9}, gives the filter characteristics of Fig. 4 in which the ultiplicative coplexity is 8% lower. Exaple 4: Setting ω c T =.9π, = 5.5, L = 5, and choosing nonlinear-phase S k (z) with N k = {7, 7, 7, 7, 7, 7} results in δ.35. The choice of nonlinear-phase S k (z), k =,,..., and linear-phase S k (z), k =,3,..., where N k = {7,,7,,7,}, 6 x 3 =[ ] 4.π.π.3π.4π.5π.6π.7π 6 x =[ ] 3 4.π.π.3π.4π.5π.6π.7π Fig. 4. Coplex error for low-delay FD filters with =.5. gives a ultiplicative coplexity which is 33% lower. The filter characteristics are shown in Fig. 5. Exaple5: If ω c T =.6π, = 4.5, L = 3, and with nonlinear-phase S k (z) in which N k = {5,5,5,5}, we get δ.4. With nonlinear-phase S k (z), k =,,..., and linear-phase S k (z), k =,3,..., where N k = {5,9,5,9}, we have a 35% lower ultiplicative coplexity. The filter characteristics are shown in Fig. 6. C. Subfilters with Different Orders As k increases, the constraints on S k (z) becoe ilder thereby allowing us to reduce the value of N k [8]. Therefore, one can solve (9) in a general for with different values of N k in different branches where soe of the nonlinear-phase subfilters can also be replaced with linear-phase subfilters. Hence, the arithetic coplexity can further be reduced. This section provides exaples to illustrate this. Exaple 6: With the specifications in Exaple 4, i.e., ω c T =.9π, = 5.5, and L = 5, we can use subfil- 3

5 .4 =[ ].4 =[ ] π.π.3π.4π.5π.6π.7π.8π.9π =[7 7 7 ] π.π.3π.4π.5π.6π.7π.8π.9π Fig. 5. Coplex error for low-delay FD filters with = x 3 =[ ] 3.π.π.3π.4π.5π.6π 4 x 3 =[ ] 3.π.π.3π.4π.5π.6π Fig. 6. Coplex error for low-delay FD filters with = 4.5. ters of different orders. Here, nonlinear-phase S k (z), k =,,..., and linear-phase S k (z), k =,3,... where N k = {7,,3,,5,3} gives the sae approxiation error. The ultiplicative coplexity is reduced by an additional 7%, as opposed to that of Fig. 5(b). The characteristics of these FD filters are shown in Fig. 7(a). Exaple 7: Here, we use the specifications in the previous exaple but S 4 (z) has an order of N 4 = 5 and it is also a Type IV linear-phase FIR filter. This shows that soe of the nonlinear-phase S k (z) can indeed be replaced with linearphase FIR filters thereby further reducing the ultiplicative coplexity. The filter characteristics are shown in Fig. 7(b). IV. CONCLUSION A ethod for designing low-delay FD filters was outlined in which both linear-phase and nonlinear-phase subfilters are used. As opposed to the case with nonlinear-phase FIR filters, in all branches of the Farrow structure, the proposed ethod uses linear-phase FIR filters in every second branch. Even π.π.3π.4π.5π.6π.7π.8π.9π =[ ].π.π.3π.4π.5π.6π.7π.8π.9π Fig. 7. Coplex error for low-delay FD filters with = 5.5. (a) Exaple 6. (b) Exaple 7. with a reduced order for these linear-phase FIR filters, the approxiation error is not affected. However, the ultiplicative coplexity is reduced by an average of 3%. The paper also showed that we can have subfilters of different orders so as to further reduce the arithetic coplexity. A fraework to systeatically obtain these orders is a topic for future work. REFERENCES [] U. Zölzer, Digital Audio Signal Processing. New York: Wiley, 997. [] W. H. T. (Editor),SoftwareDefinedRadio:EnablingTechnologies. New York: Wiley,. [3] S. R. Dooley and A. K. Nandi, On explicit tie delay estiation using the Farrow structure, Signal Processing, vol. 7, pp , 999. [4] H. Johansson and P. Löwenborg, Reconstruction of nonuniforly sapled bandliited signals by eans of digital fractional delay filters, IEEE Trans. Signal Processing, vol. 5, no., pp , Nov.. [5] J. H. Reed, Software Radio: A Modern Approach to Radio Engineering. Prentice Hall; NJ,. [6] C. W. Farrow, A continuously variable digital delay eleent, in Proc. IEEE Int. Syp. Circuits Syst., vol. 3, Espoo, Finland, June 988, pp [7] J. Vesa and T. Saraäki, Design and properties of polynoial-based fractional delay filters, in Proc. IEEE Int. Syp. Circuits Syst., Geneva, Switzerland,. [8] H. Johansson and P. Löwenborg, On the design of adjustable fractional delay FIR filters, IEEETrans.CircuitsSyst.II, vol. 5, no. 4, pp , Apr. 3. [9] J. Yli-Kaakinen and T. Saraäki, Multiplication-free polynoial-based FIR filters with an adjustable fractional delay, Circuits Syst. Signal Processing, vol. 5, no., pp , Apr. 6. [] T. I. Laakso, V. Väliaki, M. Karjalainen, and U. K. Laine, Splitting the unit delay tools for fractional delay filter design, IEEE Signal ProcessingMag., vol. 3, no., pp. 3 36, Jan [] C. K. S. Pun, Y. C. Wu, S. C. Chan, and K. L. Ho, On the design and efficient ipleentation of the Farrow structure, IEEE Signal Processing Lett., vol., no. 7, pp. 89 9, July 3. [] E. Heranowicz and H. Johansson, On designing iniax adjustable wideband fractional delay FIR filters using two-rate approach, in Proc. Eur. Conf. Circuit Theory Design, Cork, Ireland, Aug. 9-Sept. 5. [3] S. K. Mitra, Digital Signal Processing: A Coputer Based Approach. McGraw-Hill, Feb. 6. 4